Introduction

• Signals can be analyzed in both time domain and frequency domain.
• The frequency content of a signal describes how different frequency components contribute to the overall signal.
• Applications in biomedical signals, audio processing, communications, and image processing.

Convolution in Time Domain

• Convolution is a fundamental operation in signal processing.

• Given two signals ( x(t) ) and ( h(t) ), their convolution is defined as:

  \[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]

• In discrete-time, convolution is:

  \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]

Convolution Theorem

• Convolution in time domain corresponds to multiplication in frequency domain:

  \[ X(f) H(f) = Y(f) \]

• This property is crucial in filter design and system analysis.

Introduction to Fourier Series

(-1.0, 4.0)

(-1.0, 4.0)

Introduction to Fourier Series

• Convolution requiere the representation of the signal in a sum of impulse functions.

• Fourier series represents periodic signals as a sum of sinusoids:

  \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]

  where ( C_n ) are the Fourier coefficients.

• Decomposing a signal into sinusoidal components allows frequency analysis.

Fourier Coefficients

• The Fourier coefficients ( C_n ) are computed as:

  \[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]

• Determines how much of each frequency is present in the signal.

Example of Fourier Series Expansion

Example 2 of Fourier Series

(array([-5.,  0.,  5., 10., 15., 20., 25., 30., 35.]), [Text(-5.0, 0, '−5'), Text(0.0, 0, '0'), Text(5.0, 0, '5'), Text(10.0, 0, '10'), Text(15.0, 0, '15'), Text(20.0, 0, '20'), Text(25.0, 0, '25'), Text(30.0, 0, '30'), Text(35.0, 0, '35')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Example 2 of Fourier Series

(array([-2.,  0.,  2.,  4.,  6.,  8., 10., 12., 14., 16.]), [Text(-2.0, 0, '−2'), Text(0.0, 0, '0'), Text(2.0, 0, '2'), Text(4.0, 0, '4'), Text(6.0, 0, '6'), Text(8.0, 0, '8'), Text(10.0, 0, '10'), Text(12.0, 0, '12'), Text(14.0, 0, '14'), Text(16.0, 0, '16')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Relationship Between Frequency Content and Sampling Frequency

• Sampling theorem: A signal must be sampled at a frequency at least twice its highest frequency component:

  \[ f_s \geq 2 f_{max} \]

• Aliasing occurs if sampling frequency is too low.